How to show 2-tori is diffeomorphic to S^3

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  • #1
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Define 2-tori as {(z1,z2)| |z1|=c1,|z2|=c2} for c1 and c2 are constants, how to show that it is diffeomorphic to S^3
 

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  • #2
lavinia
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Define 2-tori as {(z1,z2)| |z1|=c1,|z2|=c2} for c1 and c2 are constants, how to show that it is diffeomorphic to S^3

A torus is not diffeomorphic to S^3.
 
  • #3
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My mistake, it should be how 2-tori is embedded into S^3
 
  • #4
lavinia
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My mistake, it should be how 2-tori is embedded into S^3

map R^2 into R^4 by (x,y) -> (1/2^.5)(cos x, sin x, cos y, sin y)

The image is a torus in S^3
 
  • #5
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What if there is a smooth function F:C^2\{0} to C, defined as F(z1,z2)=z1^p+z2^q with p and q>=2 and they are relatively prime, then how to show that S^3 intersect F^(-1)(0) is diffeomorphic to 2-tori?
 

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